The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$

Details The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$ The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$

2008-08-08

arxiv.org/abs/0808.1291v3

Bull. Lond. Math. Soc. 41 (2009), no. 4, 621--633

Physics

Mathematical Physics

Added reference to paper by Cohn and Kumar which shows optimality of roots of unity for -2<s<-1

Scientific paper

10.1112/blms/bdp034

We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -2$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points. By analytic continuation we deduce the expansion for all complex values of $s$. The Riemann zeta function plays an essential role in this asymptotic expansion.

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